I cannot remote desktop from one server to the next server in my network. I know the server name is correct and I have tried the IP address and neither one are working. I have restarted the server and verified that the necessary automatic services are running. The AV client firewall has never blocked any connections before. The default RDP port is still configured on the server. What else could I be missing here?

It probably has to do with the AV client and such. If you can ping the server and DNS looks good, that's where I would head, otherwise use VNC. Use UltraVNC, it's more secure than the regular VNC. At least you can set it up to use AD user and password for Ultra,

I can only access either of my 3 RDS Hosts directly by name (now that they're all part of the Farm and load balancing is working) by using the following command mstsc -v: servername /admin﻿

﻿If I use that command I can directly remote control either of the 3 by name or IP without any issue. If I try to login directly by name or IP without that command I receive the following error, 'sometimes':

"Remote desktop connection the connection cannot be completed because the remote computer that was reached is not the one you specified..."﻿

ITPT: If you are unable to restart the computer manually/physically, and it is not responding to anything you're doing, go into your Domain Controller under an elevated account, go to cmd, type shutdown /i and then add the computer you want to restart to the list. Choose whether or not you want to warn your users, add a reason, and a comment, and then click OK.